Remark 3 
Unlike [1]}, [2]}, [3]} our circuit players have identical interest both with their predecessors and their successors in the circuit. We set the utility in the interaction with predecessors higher than the interaction with predecessors. This incentivizes each circuit player to match its bit to the correct computation of the predecessors (rather than adjusting her input in a way that her successor's action will be correct). This results in an exponential decay of the utilities in the depth of the circuit. In particular, the computed potential \(\phi \)  in the output has exponentially lower weight in the common utility relative to the input.

